Although his statement addressing a chorus of criticism fell short of a full-throated apology, Mr. Zuckerberg said that Facebook would contact users whose data had been harvested through a personality quiz app and passed along to the political data firm Cambridge Analytica.

Mr. Zuckerberg, 33, was trying to quell the crisis over the disclosure last weekend that Cambridge Analytica had used data that had been improperly obtained from Facebook as the firm worked on behalf of Donald J. Trump’s presidential campaign.

“Are there other Cambridge Analyticas out there?” Mr. Zuckerberg said later in an interview with The New York Times. He added, “Were there apps which could have gotten access to more information and potentially sold it without us knowing or done something that violated people’s trust? We also need to make sure we get that under control.”

Mr. Zuckerberg said the company would investigate apps like the third-party quiz app that had previously obtained access to “large amounts of information” from the social network. He also said the company would restrict third-party developers’ access.

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In the wake of the Cambridge Analytica scandal, in which data from over 50 million Facebook profiles was secretly scraped and mined for voter insights, many Facebook users have decided to delete their accounts — but untangling yourself from a site like Facebook is not as easy as pressing “delete.”

“We also made mistakes, there’s more to do, and we need to step up and do it,” he wrote in his Facebook statement.

The Cambridge Analytica revelations added to the questions that have been raised about Facebook’s handling of user data and security. Those questions have only intensified as the company has faced criticism over the role its platform played in Russian attempts to influence the 2016 presidential election and the way it has been used to spread misinformation on the internet.

The resulting backlash is Facebook’s worst crisis since it was founded by Mr. Zuckerberg and others in 2004. The information, photos and other content that users post and their frequent engagement with the platform is crucial to the social network, and to the company’s profitability. Questions about user privacy and security threaten the company’s standing at a time when people are already uneasy about whether the use of technology can bring good or ill.

Last Friday, after The New York Times, The Observer of London and Channel 4 in Britain told Facebook that Cambridge Analytica had not deleted all of the data it had obtained, the social network banned the political consulting firm and Aleksandr Kogan, the Cambridge University researcher who created the personality quiz app that was used to harvest user data.

“This was a breach of trust between Kogan, Cambridge Analytica and Facebook,” Mr. Zuckerberg wrote on Wednesday. “But it was also a breach of trust between Facebook and the people who share their data with us and expect us to protect it.”

Facebook representatives confirmed that Cambridge Analytica representatives met with Facebook on Tuesday to discuss lifting the ban. Mr. Zuckerberg told The Times he did not rule out allowing Cambridge Analytica back, saying Facebook must first conduct a “full forensic audit of the firm” and “have full confirmation that there’s no wrongdoing here.”

The reaction to the Cambridge Analytica disclosure has been severe. Politicians in the United States and Britain have called for Mr. Zuckerberg to explain how his company handles user data, and state attorneys general in Massachusetts and New York have begun investigating Cambridge Analytica and Facebook. A #DeleteFacebook movement calling on people to close their accounts has also gathered steam.

In Washington, there have been more calls for regulation of internet companies like Facebook. Mr. Zuckerberg’s troubles there were illustrated by the final passage on Wednesday of a bill to combat sex trafficking. The bill would lift liability protections that internet companies have enjoyed for content that users post on their platforms. Facebook and other internet giants had quietly fought the bill for more than a year, but eventually dropped their opposition.

Lawmakers who have demanded that Mr. Zuckerberg testify before Congress about Facebook’s relationship with Cambridge Analytica were not appeased by his statement.

“You need to come to Congress and testify to this under oath,” Senator Ed Markey, Democrat of Massachusetts, tweeted in response to Mr. Zuckerberg’s post.

Independent researchers who have used data from Facebook said that Mr. Zuckerberg’s statement did not acknowledge how the gathering of user data was fundamental to the company’s operations.

“He avoided the big issue, which is that for many years, Facebook was basically giving away user data like it was handing out candy,” said Jonathan Albright, research director at the Tow Center for Digital Journalism at Columbia University. “There is no question that handing out that data made Facebook the success it is as a company. This has to be recognized as part of their business model and not just a one-off problem.”

In his statement, Mr. Zuckerberg laid out a timeline of Facebook’s dealings with Cambridge Analytica. He traced the information-sharing issue to 2007, when Facebook decided to become an open platform — enabling people to use Facebook to log into other apps and share detailed personal information about themselves and their friends.

In 2013, Mr. Kogan, the Cambridge researcher, created a personality quiz app that about 300,000 people installed, Mr. Zuckerberg wrote. Because Facebook was an open platform, Mr. Kogan was able to collect data on tens of millions of friends of those users who had installed the personality quiz app.

A year later, Mr. Zuckerberg said, Facebook changed its policy to limit how much data third-party apps could access. “These actions would prevent any app like Kogan’s from being able to access so much data today,” he wrote.

By 2015, Mr. Kogan had shared his data and findings with Cambridge Analytica, which later used the material to single out American voters. Mr. Zuckerberg said Facebook had banned Mr. Kogan’s app and demanded that the researcher and Cambridge Analytica formally certify that the data had been deleted. He did not address in his post why Facebook had not already disclosed those activities to its users whose data had been harvested by Mr. Kogan and Cambridge Analytica.

“Whenever there’s an issue where someone’s data gets passed to someone who the rules of the system shouldn’t have allowed it to, that’s rightfully a big issue and deserves to be a big uproar,” Mr. Zuckerberg said in the interview.

For Mr. Zuckerberg, the outcry over Cambridge Analytica has been personally damaging. Inside Facebook, even his staunch supporters have described a tense atmosphere. Some employees have sought to transfer to other divisions, such as the messaging app WhatsApp and the photo-sharing platform Instagram, calling their work on Facebook’s main product “demoralizing.”

Mr. Zuckerberg spent part of the past week hunkered down with a small group of engineers to discuss how to make information on Facebook’s users more secure, and to potentially give them more control of their data, according to two Facebook employees who declined to be named because the proceedings were confidential.

His silence on the matter has prompted mounting criticism in the past few days. Facebook held a staff meeting on Tuesday to answer questions about Cambridge Analytica and the surrounding outcry, but Mr. Zuckerberg did not appear at the event. He was scheduled to appear at a staff meeting that was set for Friday.

In his interview with The Times, Mr. Zuckerberg said that the company’s efforts to safeguard its platform from bad behavior — which includes preparing for possible interference attempts in the 2018 midterm elections — were an important part of a larger transformation at the company, which has had to adjust from its roots as a social network for college students into a powerful global information hub.

“If you had asked me, when I got started with Facebook, if one of the central things I’d need to work on now is preventing governments from interfering in each other’s elections, there’s no way I thought that’s what I’d be doing if we talked in 2004 in my dorm room,” Mr. Zuckerberg said.